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Bit-level systolic arrays for modular multiplication

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Abstract

This paper presents bit-level cellular arrays implementing Blakley's algorithm for multiplication of twon-bit integers modulo anothern-bit integer. The semi-systolic version uses 3n(n+3) single-bit carry save adders and 2n copies of 3-bit carry look-ahead logic, and computes a pair of binary numbers (C, S) in 3n clock cycles such thatC+Sε[0, 2N). The carry look-ahead logic is used to estimate the sign of the partial product, which is needed during the reduction process. The final result in the correct range [0,N) can easily be obtained by computingC+S andC+S−N, and selecting the latter if it is positive; otherwise, the former is selected. We construct a localized process dependence graph of this algorithm, and introduce a systolic array containing 3nw simple adder cells. The latency of the systolic array is 6n+w−2, wherew=⌈n/2⌉. The systolic version does not require broadcast and can be used to efficiently compute several modular multiplications in a pipelined fashion, producing a result in every clock cycle.

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Koç, Ç.K., Hung, C.Y. Bit-level systolic arrays for modular multiplication. J VLSI Sign Process Syst Sign Image Video Technol 3, 215–223 (1991). https://doi.org/10.1007/BF00925832

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Key Words

  • modular multiplication
  • carry save adders
  • sign estimation
  • systolic array
  • scheduling